Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 340 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.Which of these numbers is divisible by 9? (a) 5069 (b) 5082 (c) 5067 (d) 5063
2. 2.Which of these numbers is divisible by 6? (a) 4982 (b) 4970 (c) 4974 (d) 4977
3. 3.What is the units digit of 3^35? (a) 9 (b) 3 (c) 7 (d) 1
4. 4.What is the units digit of 3^26? (a) 1 (b) 9 (c) 3 (d) 7
5. 5.What is the smallest number you must add to 132 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
6. 6.Which of these numbers is divisible by 3? (a) 223 (b) 222 (c) 221 (d) 226
7. 7.What is the remainder when 152 is divided by 3? (a) 2 (b) 0 (c) 1
8. 8.Which of these numbers is divisible by 6? (a) 4121 (b) 4122 (c) 4133 (d) 4123
9. 9.What is the remainder when 249 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
10. 10.What is the remainder when 278 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
11. 11.Which of these numbers is divisible by 8? (a) 7067 (b) 7086 (c) 7072 (d) 7079
12. 12.What is the remainder when 98 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
13. 13.What is the units digit of 9^48? (a) 9 (b) 0 (c) 2 (d) 1
14. 14.What is the units digit of 3^60? (a) 3 (b) 7 (c) 9 (d) 1
15. 15.What is the units digit of 4^40? (a) 4 (b) 7 (c) 6 (d) 5
16. 16.What is the smallest number you must add to 627 to make it divisible by 9? (a) 4 (b) 2 (c) 3 (d) 6
17. 17.What is the smallest number you must add to 367 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
18. 18.Which of these numbers is divisible by 11? (a) 5696 (b) 5684 (c) 5704 (d) 5687
19. 19.What is the smallest number you must add to 515 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
20. 20.What is the remainder when 402 is divided by 8? (a) 6 (b) 1 (c) 2 (d) 3
21. 21.What is the smallest number you must add to 198 to make it divisible by 10? (a) 2 (b) 1 (c) 8 (d) 3
22. 22.What is the remainder when 253 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
23. 23.What is the smallest number you must add to 286 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
24. 24.What is the smallest number you must add to 235 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
25. 25.Which of these numbers is divisible by 4? (a) 2534 (b) 2536 (c) 2542 (d) 2537
26. 26.What is the smallest number you must add to 707 to make it divisible by 11? (a) 3 (b) 7 (c) 9 (d) 8
27. 27.What is the smallest number you must add to 126 to make it divisible by 4? (a) 1 (b) 2 (c) 3
28. 28.What is the units digit of 4^56? (a) 7 (b) 6 (c) 5 (d) 4
29. 29.What is the units digit of 4^51? (a) 4 (b) 3 (c) 5 (d) 6
30. 30.What is the smallest number you must add to 61 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
31. 31.What is the units digit of 2^52? (a) 8 (b) 2 (c) 4 (d) 6
32. 32.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
33. 33.Which of these numbers is divisible by 9? (a) 6539 (b) 6557 (c) 6544 (d) 6543
34. 34.What is the smallest number you must add to 477 to make it divisible by 12? (a) 2 (b) 4 (c) 3 (d) 9
35. 35.Which of these numbers is divisible by 8? (a) 3676 (b) 3687 (c) 3692 (d) 3680
36. 36.What is the remainder when 62 is divided by 3? (a) 2 (b) 0 (c) 1
37. 37.What is the units digit of 8^56? (a) 4 (b) 8 (c) 2 (d) 6
38. 38.What is the units digit of 2^28? (a) 6 (b) 8 (c) 4 (d) 2
39. 39.What is the remainder when 122 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
40. 40.What is the remainder when 237 is divided by 9? (a) 3 (b) 2 (c) 6 (d) 4

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